It is currently Wed Dec 13, 2017 2:17 am


All times are UTC [ DST ]




Post new topic Reply to topic  [ 1 post ] 
Author Message
PostPosted: Thu Jul 14, 2016 1:03 pm 
Administrator
Administrator
User avatar

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 806
Location: Larisa
An other way to define the trigonometric functions is by using their power series, that is: $$ \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!},\; \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{\left ( 2n \right )!}, \; x \in \mathbb{R} $$ The classic definition of the trigonometric functions is based on the unit circle.

a. Use the definition given above to prove that:
  1. \( \sin 0 =0, \; \cos 0 =1 \)
  2. \( \left ( \sin x \right )'= \cos x , \; \left ( \cos x \right )' =-\sin x \)
  3. \( \sin^2 x + \cos^2 x =1 \)

b. Prove that the classic definition, the definition given above and the definition \( \displaystyle \sin x = \frac{e^{ix}-e^{-ix}}{2i}, \; \cos x =\frac{e^{ix}+e^{-ix}}{2} \) are equivalent.

_________________
Imagination is much more important than knowledge.
Image


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 1 post ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
cron
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net